Understanding Automotive Electronics 8th - Appendix B

APPENDIX

DISCRETE TIME SYSTEMS THEORY

B

As explained in Appendix A, automotive electronic control and instrumentation systems (as well as

virtually all other electrical systems) are implemented with digital electronics as at least some com-

ponent or subsystem. Digital controllers and/or signal processing subsystems incorporate one or more

microprocessors or microcontrollers, each having a stored program to run the system. Su ch systems are

fundamentally discrete time systems.

However, automotive electronic systems also incorporate analog or continuous time components

Although it is not necessary, most discrete time systems use periodic times to represent input/out-

put; that is, the kth discrete time is given by

t

k

where T

system and an analog destination component is depicted in Fig. B.1.

In this figure, the source has a continuous time electrical signal v(t) that could, for example, be a

sensor output. The interface electronics-labeled A/D converter (which is modeled later in this appendix)

generates a sequence of numerical values called samples at each discrete time or “sample period” t

k

is denoted x

k

, which is a digital (N bit) numerical value equal to v

k

becomes a binary number x

(as well as previous samples depending on the operations performed). Although the destination

component might be a display device that can display the desired output numerical value, it may also

be an actuator requiring a continuous time electrical signal y(t). We assume here that the destination

component (e.g., a display or actuator) requires a continuous time electrical input. This continuous time

electrical signal is generated from the output y

concerned with specific auto motive systems, we deal with nonlinearities as required to explain the

particular system.

As shown in the Appe ndix A, a linear time-invariant continuous time system is characterized by an

nth order differential equation with constant coefficients. The linear time-invariant discrete time sys-

tem is characterized by a model in the format of difference equations. One commonly used model for

calculating the output y

n

of a discrete time system is in the form of a recursive model:

y

a

b

y

k

‘

It is shown in Appendix A that a continuous time system is usefully characterized by its transfer

system (H(s)), whi ch is obtained from the Laplace transforms of its input and output. A similar pro-

cedure is very useful for conducting performance analysis or design of a discrete time digital system.

For a discrete time system, the transform of a sequence x

n

for a continuous time system is the Z-transform X(z) defined by

We illustrate with an example in which x

n

is defined as

x

n

n

The Z-transform is given by

∞

where z ¼ complex variable analogous to s for the Laplace transform. This latter sum is a geometric

series that conver ges if cz

There are several important elementary properties of the Z-transform that are important in the analysis

of discrete time digital systems that are summarized without proof below:

n

n

k

k

∞

x

y

k

then

linear time invariance for this component, it has already been explained that its model is generally of

the recursive form

y

K

b

k

y

Such a subsystem is typically called a digital filter, regardless of its specific function in the larger sys-

tem. We proceed with the approach to the design/analysis of the digital filter by first determining its

digital transfer function. This can be computed directly from the Z-transform of the above model:

∞

k

x

L

b

k

y

k

which can be rewritten in the form

The function H(z) is the digital transfer function of the digital filter and is given by

a

z

b

The design procedures presented later in this appendix permit the calculation of the digital transfer

inverse Z-transform of H(z) is the sequence {h

n

}, where components are given by

h

n

Limh

n

the filter is assured to be stable.

A filter that has all b

yield the most recent output y

. Such a filter is also said to have a finite impulse response since

One of the most important inputs for assessing system performance is the sinusoid. For an understanding

of the sinusoidal frequency response of a digital filter, it is necessary to have the Z-transform of a sam-

pled sinusoidal signal having frequency ω sampled at period t

n

where Ω ¼ ωT. The sinusoid can be rewritten as

∞

n

∞

n

z